Graphics Reference
In-Depth Information
One way to overcome this problem is to use the pseudo-inverse of
w
. A common approach
to pseudo-inverse is to decompose the covariance matrix using SVD,
U
and then
invert the non-zero singular values (or only those that are considerably higher than zero and
the remaining ones are set to zeros),
w
=
U
w
=
+
U
, where
+
is diagonal and each diagonal
U
element is the inverse (1
if it is non-zero. A more
accepted approach is to reduce the dimensionality of the training data using PCA prior to the
application of LDA, called PCA
/λ
i
) of the corresponding element in
LDA.
LDA is widely used in both 2D and 3D face recognition and generally demonstrates better
recognition accuracies than does PCA. Apart from its relative discriminatory advantage, LDA
still suffers from pose and expression variations. There are three prominent approaches to
LDA feature extraction in 3D face recognition. First, LDA feature extraction is performed
on range images. Second, LDA features are extracted from facial normal maps. Gokberk
et al. (2006) experimentally have shown that LDA on normal maps outperforms that on range
images. Third, in textured 3D face recognition, LDA feature extraction has been performed
on the texture, whereas ICP, which is more accurate than 3D LDA, is used for shape feature
extraction (Lu et al., 2006). In matching, the dissimilarity measures produced by the shape
and texture are combined using a weighted sum rule.
+
Curvature-based Methods
In comparison to the use of curvatures for regional and point-based feature extraction, 3D face
detection and pose estimation, the use of curvatures in holistic 3D face recognition is to an
extent limited. However, some of the most early approaches to 3D face recognition are those
based on the extended Gaussian image (EGI) (Lee and Milios, 1990; Tanaka et al., 1998).
EGI facial features:
An EGI is constructed by mapping (or associating) the facial surface
normals to those of a unit sphere in the same directions. The points on the unit sphere
are assigned the values of the Gaussian curvatures
K
at their corresponding surface points,
G
(
K
i
, but those with no correspondences are assigned a value of zero. In reality, a
discretization of the unit sphere is used, in which some cells are within the continuous range of
normals but are left empty due to the sampling of the surface. Interpolation is used to estimate
the curvature at those cells. In the EGI variant by Tanaka et al. (1998), the facial surface is
segmented into valleys and ridges based on
H
and
K
curvatures. EGIs are then computed
from the two facial segments. In that work, the EGIs map the principal curvature directions
(
e
1
and
e
2
) to the normals of a unit sphere rather than from the normals of the original surface,
to the normals of a unit shpere.
Surface representation using EGI has some advantages. Since the curvatures are invari-
ant to rigid transformations and the normals are invariant to translations, EGI representa-
tions/features are also invariant to translations. However, the rotational invariance is easily
handled during matching by means of spherical correlation, in which one of the spheres
of the two matched EG
Is,
G
1
(
θ
i
,φ
i
)
=
θ,φ
) and
G
2
(
θ,φ
), is revoluted to maximize the correlation
G
1
(
G
1
(
)
G
2
(
).
Ideally, when a 3D surface represented by an EGI is convex, there is a one-to-one mapping
between the surface and the unit sphere normals as there are no surface normals pointing in
the same direction. This makes EGI representations for convex surfaces unique. However,
θ,φ
)
G
2
(
θ,φ
)
/
θ,φ
θ,φ
